Properties

Label 5070.2.a.bz
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.131472.2
Defining polynomial: \( x^{4} - 2x^{3} - 19x^{2} + 20x + 52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - \beta_1 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - \beta_1 q^{7} - q^{8} + q^{9} - q^{10} + (\beta_{3} + 1) q^{11} + q^{12} + \beta_1 q^{14} + q^{15} + q^{16} + 4 q^{17} - q^{18} + ( - \beta_{3} - \beta_1) q^{19} + q^{20} - \beta_1 q^{21} + ( - \beta_{3} - 1) q^{22} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{23} - q^{24} + q^{25} + q^{27} - \beta_1 q^{28} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{29} - q^{30} + (\beta_{3} - 3 \beta_{2} - \beta_1) q^{31} - q^{32} + (\beta_{3} + 1) q^{33} - 4 q^{34} - \beta_1 q^{35} + q^{36} + ( - \beta_{3} + 3 \beta_{2} + 2) q^{37} + (\beta_{3} + \beta_1) q^{38} - q^{40} + (2 \beta_1 - 2) q^{41} + \beta_1 q^{42} + ( - 4 \beta_{2} + \beta_1 + 3) q^{43} + (\beta_{3} + 1) q^{44} + q^{45} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{46} + (2 \beta_{2} + 2 \beta_1 - 3) q^{47} + q^{48} + (4 \beta_{2} + \beta_1 + 3) q^{49} - q^{50} + 4 q^{51} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{53} - q^{54} + (\beta_{3} + 1) q^{55} + \beta_1 q^{56} + ( - \beta_{3} - \beta_1) q^{57} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{58} + ( - 2 \beta_{2} + \beta_1 + 1) q^{59} + q^{60} + (2 \beta_{2} + 4) q^{61} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{62} - \beta_1 q^{63} + q^{64} + ( - \beta_{3} - 1) q^{66} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{67} + 4 q^{68} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{69} + \beta_1 q^{70} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{71} - q^{72} + (4 \beta_{2} - 2 \beta_1 - 2) q^{73} + (\beta_{3} - 3 \beta_{2} - 2) q^{74} + q^{75} + ( - \beta_{3} - \beta_1) q^{76} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 - 2) q^{77} + ( - 3 \beta_1 - 1) q^{79} + q^{80} + q^{81} + ( - 2 \beta_1 + 2) q^{82} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{83} - \beta_1 q^{84} + 4 q^{85} + (4 \beta_{2} - \beta_1 - 3) q^{86} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{87} + ( - \beta_{3} - 1) q^{88} + ( - \beta_{3} + 4 \beta_{2} + \beta_1) q^{89} - q^{90} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{92} + (\beta_{3} - 3 \beta_{2} - \beta_1) q^{93} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{94} + ( - \beta_{3} - \beta_1) q^{95} - q^{96} + ( - 2 \beta_{2} + 2 \beta_1 + 8) q^{97} + ( - 4 \beta_{2} - \beta_1 - 3) q^{98} + (\beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{15} + 4 q^{16} + 16 q^{17} - 4 q^{18} + 4 q^{20} - 2 q^{21} - 2 q^{22} + 4 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27} - 2 q^{28} + 8 q^{29} - 4 q^{30} - 4 q^{31} - 4 q^{32} + 2 q^{33} - 16 q^{34} - 2 q^{35} + 4 q^{36} + 10 q^{37} - 4 q^{40} - 4 q^{41} + 2 q^{42} + 14 q^{43} + 2 q^{44} + 4 q^{45} - 4 q^{46} - 8 q^{47} + 4 q^{48} + 14 q^{49} - 4 q^{50} + 16 q^{51} + 8 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{56} - 8 q^{58} + 6 q^{59} + 4 q^{60} + 16 q^{61} + 4 q^{62} - 2 q^{63} + 4 q^{64} - 2 q^{66} - 12 q^{67} + 16 q^{68} + 4 q^{69} + 2 q^{70} - 16 q^{71} - 4 q^{72} - 12 q^{73} - 10 q^{74} + 4 q^{75} - 8 q^{77} - 10 q^{79} + 4 q^{80} + 4 q^{81} + 4 q^{82} - 16 q^{83} - 2 q^{84} + 16 q^{85} - 14 q^{86} + 8 q^{87} - 2 q^{88} + 4 q^{89} - 4 q^{90} + 4 q^{92} - 4 q^{93} + 8 q^{94} - 4 q^{96} + 36 q^{97} - 14 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 19x^{2} + 20x + 52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} - 14\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4\beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} + 4\beta_{2} + 15\beta _1 + 10 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.64466
2.32258
−1.32258
−3.64466
−1.00000 1.00000 1.00000 1.00000 −1.00000 −4.64466 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −2.32258 −1.00000 1.00000 −1.00000
1.3 −1.00000 1.00000 1.00000 1.00000 −1.00000 1.32258 −1.00000 1.00000 −1.00000
1.4 −1.00000 1.00000 1.00000 1.00000 −1.00000 3.64466 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.a.bz 4
13.b even 2 1 5070.2.a.ca 4
13.d odd 4 2 5070.2.b.ba 8
13.f odd 12 2 390.2.bb.c 8
39.k even 12 2 1170.2.bs.f 8
65.o even 12 2 1950.2.y.k 8
65.s odd 12 2 1950.2.bc.g 8
65.t even 12 2 1950.2.y.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 13.f odd 12 2
1170.2.bs.f 8 39.k even 12 2
1950.2.y.j 8 65.t even 12 2
1950.2.y.k 8 65.o even 12 2
1950.2.bc.g 8 65.s odd 12 2
5070.2.a.bz 4 1.a even 1 1 trivial
5070.2.a.ca 4 13.b even 2 1
5070.2.b.ba 8 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{4} + 2T_{7}^{3} - 19T_{7}^{2} - 20T_{7} + 52 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 34T_{11}^{2} + 62T_{11} + 181 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display
\( T_{31}^{4} + 4T_{31}^{3} - 79T_{31}^{2} - 556T_{31} - 956 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 19 T^{2} - 20 T + 52 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} - 34 T^{2} + 62 T + 181 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T - 4)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 63 T^{2} + 72 T + 468 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 43 T^{2} + 16 T + 52 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} - 39 T^{2} + 148 T + 376 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} - 79 T^{2} - 556 T - 956 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} - 34 T^{2} + \cdots - 803 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} - 76 T^{2} - 160 T + 832 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} - 43 T^{2} + \cdots - 572 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} - 82 T^{2} + \cdots - 1103 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} - 75 T^{2} + 4 T + 52 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 31 T^{2} + 216 T - 284 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 52 T^{2} + \cdots + 352 \) Copy content Toggle raw display
$71$ \( T^{4} + 16 T^{3} - 100 T^{2} + \cdots - 5024 \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} - 124 T^{2} + \cdots - 4544 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} - 147 T^{2} + \cdots + 3508 \) Copy content Toggle raw display
$83$ \( T^{4} + 16 T^{3} - 100 T^{2} + \cdots - 5024 \) Copy content Toggle raw display
$89$ \( T^{4} - 4 T^{3} - 115 T^{2} + \cdots - 1004 \) Copy content Toggle raw display
$97$ \( T^{4} - 36 T^{3} + 380 T^{2} + \cdots - 5408 \) Copy content Toggle raw display
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